Quantum Circuits for Exact Unitary t-Designs and Applications to Higher-Order Randomized Benchmarking

A unitary t-design is a powerful tool in quantum information science and fundamental physics. Despite its usefulness, only approximate implementations were known for general t. In this paper, we provide quantum circuits that generate exact unitary t-designs for any t on an arbitrary number of qubits. Our construction is inductive and is of practical use in small systems. We then introduce a tth-order generalization of randomized benchmarking (t-RB) as an application of exact 2t-designs. We particularly study the 2-RB in detail and show that it reveals self-adjointness of quantum noise, a metric related to the feasibility of quantum error correction (QEC). We numerically demonstrate that the 2-RB in one-and two-qubit systems is feasible, and experimentally characterize background noise of a superconducting qubit by the 2-RB. It is shown from the experiment that interactions with adjacent qubits induce the noise that may result in an obstacle toward a realization of QEC.